Vertex Group Weight Influence
The location that you have the issue is the barrier between what gets moved and what doesn't. The mesh faces in this section can easily get stretched too much with the motion of the attached pieces.

I would recommend using a curve object for the hose instead of a mesh. By giving the curve a full fill and a bevel depth and increasing the resolution you get a nice round hose.



To animate the hose you add a hook to one or more of the curve points. While in edit mode select one of the curve points and press CtrlH and select Hook to New Object. This will create an empty at the location of the curve point as well as add a hook constraint. in object mode when you move the empty the curve point will move with it.



The beauty of a curve in this scenario is it maintains the hose shape no matter how far you stretch the points from one another. Parenting the empty to your piston will keep the end of the hose in place.

while stretching the hose too far will create a noticeable distortion you may want to add more hooks to the next curve point and use a limit distance constraint so that the coiled hose picks up and follows the movement.



For the final animation you would want to animate some rotation on the empties to prevent kinks in the hose.

You can look at my simple example file here.